Nearly integral homomorphisms of commutative rings

Abstract

A unital homomorphism f: R → T of commutative rings is said to be nearly integral if the induced map R/I → T/IT is integral for each ideal I of R which properly contains ker (f). This concept leads to new characterisations of integral extensions and fields. For instance, if R is not a field, then an inclusion R → T is integral if and only if it is nearly integral and (R, T) is a lying-over pair. It is also proved that each overring extension of an integral domain R is nearly integral if and only if dim (R) ≤ 1 and the integral closure of R is a Prüfer domain. Related properties and examples are also studied.